# The Conjugate Main Eigenvalues Of Vertex Deleted Subgraphs of a Strongly Regular Graph

### From NZJM

**New Zealand Journal of Mathematics**

**Vol.** 40, (2010), **Pages** 67-74

**Mirko LepoviÄ‡ **

Tihomira VuksanoviÄ‡a 32,

34000, Kragujevac,

Serbia

**Abstract** Let *G* be a simple graph of order *n*. Let and
, where *a* and *b* are two nonzero
integers and *m* is a positive integer such that *m* is not a
perfect square. We say that *A*^{c} = [*c*_{ij}] is the conjugate
adjacency matrix of the graph *G* if *c*_{ij} = *c* for any two
adjacent vertices *i* and *j*, for any two
nonadjacent vertices *i* and *j*, and *c*_{ij} = 0 if *i* = *j*. Let
for any nonnegative integer *k*. Further,
let *G* be a strongly regular graph of degree ,
understanding that *G* is not the complete graph. Then for any two adjacent vertices *i* and *j* and
for any two distinct nonadjacent vertices
*i* and *j*, where τ^{c} and θ^{c} are two fixed real
numbers. Let . We demonstrate
that

where and are the conjugate main eigenvalues of
its vertex deleted subgraphs for *i* = 1,2,...,*n*.

**Keywords** strongly regular graph, conjugate adjacency matrix, conjugate main eigenvalue.

**Classification** (MSC2000) 05C50